Optimized hardward architecture and method for ecc point addition using mixed affine-jacobian coordinates over short weierstrass curves

ABSTRACT

An optimized hardware architecture and method introducing a simple arithmetic processor that allows efficient implementation of an Elliptic Curve Cryptography point addition algorithm for mixed Affine-Jacobian coordinates. The optimized architecture additionally reduces the required storage for intermediate values.

BACKGROUND

Electronic devices are becoming a ubiquitous part of everyday life. The number of smartphones and personal tablet computers in use is rapidly growing. A side effect of the increasing use of smartphones and personal tablets is that increasingly the device are used for storing confidential data such as personal and banking data. Protection of this data against theft is of paramount importance.

The field of cryptography offers protection tools for keeping this confidential data safe. Based on hard to solve mathematical problems, cryptography typically requires highly computationally intensive calculations that are the main barrier to wider application in cloud and ubiquitous computing (ubicomp). If cryptographic operations cannot be performed quickly enough, cryptography tools are typically not accepted for use on the Internet. In order to be transparent while still providing security and data integrity, cryptographic tools need to follow trends driven by the need for high speed and the low power consumption needed in mobile applications.

Public key algorithms are typically the most computationally intensive calculations in cryptography. For example, take the case of Elliptic Curve Cryptography (ECC), one of the most computationally efficient public key algorithms. The 256 bit version of ECC provides security that is equivalent to a 128 bit symmetric key. A 256 bit ECC public key should provide comparable security to a 3072 bit RSA public key. The fundamental operation of ECC is a point multiplication which is an operation heavily based on modular multiplication, i.e. approximately 3500 modular multiplications of 256 bit integers are needed for performing one ECC 256 point multiplication. Higher security levels (larger bit integers) require even more computational effort.

Building an efficient implementation of ECC is typically non-trivial and involves multiple stages. FIG. 1 illustrates stages 101, 102 and 103 that are needed to realize the Elliptical Curve Digital Signature Algorithm (ECDSA), which is one of the applications of ECC. Stage 101 deals with finite field arithmetic that comprises modular addition, inversion and multiplication. Stage 102 deals with point addition and point doubling which comprises the Joint Sparse Form (JSF), Non-Adjacent Form (NAF), windowing and projective coordinates. Finally, stage 103 deals with the ECDSA and the acceptance or rejection of the digital signature.

Any elliptic curve can be written as a plane geometric curve defined by the equation of the form (assuming the characteristic of the coefficient field is not equal to 2 or 3):

y ² =x ³ +ax+b  (1)

that is non-singular; that is it has no cusps or self-intersections and is known as the short Weierstrass form where a and b are integers. The case where a=−3 is typically used in several standards such as those published by NIST, SEC and ANSI which makes this the case of typical interest.

Many algorithms have been proposed in the literature for efficient implementation of the Point Addition (PADD) and Point Doubling (PDBL) operations. Many of these algorithms are optimized for software implementation. While these are typically efficient on certain platforms, the algorithms are typically not optimal once the underlying hardware can be tailored to the algorithm.

A PADD algorithm for mixed affine-Jacobian coordinates has been described by Cohen, Miyaji and Ono in Proceedings of the International Conference on the Theory and Applications of Cryptography and Information Security; Advances in Cryptology, ASIACRYPT 1998, pages 51-65, Springer-Verlag, 1998. Jacobian coordinates are projective coordinates where each point is represented as three coordinates (X, Y, Z) where x=X/Z², y=Y/Z³ and affine coordinates are the familiar (x,y) coordinates. Note the coordinates are all integers. PADD algorithm 200 requires 8 modular multiplications, 3 modular squarings, 6 modular subtractions, and one modular multiplication by 2 and is shown in FIG. 2. In order to perform the PADD, the algorithm further requires a minimum of 4 temporary registers, which for ECC 256 bit each need to be 256 bits in size. All operations are done in the finite field K over which the elliptic curve E is defined. The finite arithmetic field K is defined over the prime number p so that all arithmetic operations are performed modulo p. The additive identity element is the point at infinity.

SUMMARY

An optimized hardware architecture and method reduces storage requirements and speeds up the execution of the ECC PADD algorithm by requiring only two temporary storage registers and by introducing a simple arithmetic unit for performing modular subtraction and modular multiplication by 2.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows stages 101, 102 and 103 that are needed to realize the Elliptical Curve Digital Signature Algorithm (ECDSA).

FIG. 2 shows a prior art point addition algorithm.

FIG. 3 shows an embodiment in accordance with the invention.

FIG. 4 show an embodiment in accordance with the invention.

FIG. 5 shows an embodiment in accordance with the invention.

FIG. 6 shows an embodiment in accordance with the invention.

FIG. 7 shows an embodiment in accordance with the invention.

DETAILED DESCRIPTION

PADD algorithm 300 in accordance with the invention is shown in FIG. 3. PADD algorithm 300 requires fewer steps and reduces the storage requirements compared to PADD algorithm 200 for the same modular addition of two points. PADD algorithm 300 requires only two temporary storage registers, T₁ and T₂. Note, PADD algorithm 300 performs modular point addition using mixed affine-Jacobian coordinates to avoid the need for a modular inversion operation that is typically one to two orders of magnitude slower than a modular multiplication operation. The use of mixed coordinates provides a speed advantage over performing the point addition solely in Jacobian coordinates that also obviates the need for a modular inversion operation. PADD algorithm 300 is implemented over an optimized hardware architecture shown in FIG. 6 and FIG. 7 and specifically designed to take advantage of PADD algorithm 300.

As input in step 301, PADD algorithm 300 shown in FIG. 3 takes point P=(X₁, Y₁, Z₁) in Jacobian coordinates and point Q=(x₂, y₂) in affine coordinates as the two points to be added together as P+Q. T₁ and T₂ are temporary storage variables. Note that all mathematical operations shown are in modular arithmetic. In step 302 of PADD algorithm 300, the value of point P is returned as the result of the modular addition of P+Q if Q=∞, as a point at infinity is the identity element. Similarly, in step 303, the value of point Q is returned as the result of the modular addition of P+Q if P=∞, as a point at infinity is the additive identity element. In step 304, the Jacobian coordinate Z₁ is squared and the resulting value stored in temporary register T₁. In step 305, Z₁*T₁ is calculated and the resulting value stored in temporary register T₂. In step 306, T₂*y₂−Y₁ is calculated, where y₂ is in affine coordinates and Y₁ is in Jacobian coordinates, the result being stored in temporary register T₂. In step 307, the value stored in temporary register T₁ is multiplied by x₂ and X₁ is then subtracted from the result, where x₂ is in affine coordinates and X₁ is in Jacobian coordinates, the result being stored in temporary register T₁. Step 308 provides for a return if T₁ and T₂ are both zero as this means P=Q and step 309 provides for a return if T₁ is zero and T₂ is not zero as this means P=−Q. In step 310, the Jacobian coordinate Z₁ is multiplied by the value in temporary register T₁ and the result is stored as Jacobian coordinate Z₃. In step 311, the value stored in temporary register T₁ is squared and stored as Jacobian coordinate Y₃. In step 312, the value stored in temporary register T₂ is squared and stored as Jacobian coordinate X₃. In step 313, Y₃*T₁ is calculated and the result is stored in temporary register T₁. In step 314, T₁+2Y₃*X₁ is calculated and subtracted from Jacobian coordinate X₃ with the result stored as Jacobian coordinate X₃. In step 315, Y₃*X₁−X₃ is calculated and multiplied by T₂ and stored as Jacobian coordinate Y₃. Note that Y₃*X₁ was calculated in step 314 and that value is used in step 315 and is not calculated again in step 315. In step 316, T₁*Y₁ is calculated and subtracted from Jacobian coordinate Y₃ and the result is stored as Jacobian coordinate Y₃. Finally, in step 317 the result of the point addition of P+Q: (X₃, Y₃, Z₃) is returned in Jacobian coordinates.

The most computationally intensive operation in PADD algorithm 300 in FIG. 3 is modular multiplication denoted by “*”. Because most of the steps described in PADD algorithm 300 depend on the previous steps of the algorithm, it is typically most efficient to implement PADD algorithm 300 in hardware using a single modular multiplier although more than one modular multiplier may be used in accordance with the invention. Using only one modular multiplier restricts each step in PADD algorithm 300 to having no more than one modular multiplication. While step 315 appears to contain two modular multiplications, the result of Y₃*X₁ has already been calculated in step 314 and is fed in directly into the input of the hardware modular multiplier.

It is important to note that besides the modular multiplication steps performed in steps 306, 307, 314, 315 and 316 of PADD algorithm 300, two additional, comparatively simple operations are performed as well: modular subtraction and modular multiplication by 2. Note that multiplication or division by a power of 2 in binary is merely a shift operation. In order to speed up execution of PADD algorithm 300 and eliminate the need for additional temporary registers, an embodiment in accordance with the invention of simple arithmetic unit (SAU) 400 with the inputs and outputs as shown in FIG. 4.

FIG. 5 shows how steps 306, 307, 314, 315 and 316 of PADD 300 in FIG. 3 are broken down for utilization of SAU 400 which has inputs A, B and C with output D. Note that the input and output labels of SAU 400 correspond to the respective variable names in FIG. 5. Block 501 shows how step 306 of PADD algorithm 300 is broken down using SAU 400 and involves setting inputs A=T₂*y₂ and B=Y₁ with output D=A−B. Output D is written to temporary register T₂. Block 501 shows how step 307 of PADD algorithm 300 is broken down using SAU 400 and involves setting inputs A=T₁*x₂ and B=X₁ with output D=A−B. Output D is then written to temporary register T₁. Block 503 shows how step 314 of PADD algorithm 300 is broken down using SAU 400 and involves setting inputs A=X₃, B=T₁, C=y₂*X₁ with output D=A−B−2C. Output D is written to Jacobian coordinate X₃. Block 504 shows how step 315 of PADD algorithm 300 is broken down using SAU 400 and involves setting inputs A=Y₃*X₁ and B=X₃ with output D=A−B. Output D is written to Jacobian coordinate Y₃. Block 505 shows how step 316 of PADD algorithm 300 is broken down using SAU 400 and involves setting inputs A=Y₃ and B=T₁*Y₁ with output D=A−B. Output D is written to Jacobian coordinate Y₃. Note that “don't care” indicates the value is irrelevant to the calculation being performed in the respective steps.

FIG. 6 shows embodiment 600 in accordance with the invention comprising multi-cycle multiplier 610 with output register (not shown), SAU 400, multiplexer (MUX) 620 and MUX 630 with input registers X₁, Y₁, Z₁, x₂, y₂, output registers X₃, Y₃, Z₃ and temporary registers T₁ and T₂ that are all part of register memory 695. Note the individual register labels correspond to variable names in FIGS. 3 and 5. MUX 620, 630 and 740 (part of SAU 400, see FIG. 7) are controlled by the microprocessor (not shown) which executes PADD algorithm 300. As noted above, each step in PADD algorithm 300 involve at most one modular multiplication (not counting multiplication or division by 2 which in binary representation is merely a shift operation).

SAU 400 shown in FIG. 7 comprises subtractors 710 and 720, logical one bit left shifter 715 and MUX 720. Input A connects to the minuend input of subtractor 710 on line 670 and input B connects to the subtrahend input of subtractor 710 on line 675. Input C connects to logical one bit left shifter 715 on line 650 where logical one bit left shifter 715 performs a multiplication of the input C by two. Subtractor 710 outputs A−B on line 730 which connects to the minuend input of subtractor 720 and the “0” input for MUX 740. Logical one bit left shifter 715 outputs 2C on line 735 to the subtrahend input of subtractor 720. Subtractor 720 outputs A−B−2C on line 750 to the “1” input for MUX 740. MUX 740 sends D on line 690.

Multi-cycle multiplier 610 functions by multiplying the values on inputs 635 and 640 together and outputting the result. Steps 301-303 are performed in the microprocessor (not shown) without using multi-cycle multiplier 610 and SAU 400.

Step 304 utilizes multi-cycle multiplier 610. Register memory 695 provides Z₁ on both inputs 635 and 640 of multi-cycle multiplier 610 and multi-cycle multiplier 610 computes Z₁ ² which is sent on line 650 to register memory 695 and stored in temporary register T₁.

Step 305 utilizes multi-cycle multiplier 610. Register memory 695 provides T₁ on input 635 and Z₁ on input 640 of multi-cycle multiplier 610. Multi-cycle multiplier 610 computes T₁*Z₁ which is sent on line 650 to register memory 695 where it is stored in temporary register T₂.

Step 306 utilizes both multi-cycle multiplier 610 and SAU 400. Register memory 695 provides T₂ and y₂ on lines 635 and 640, respectively, to multi-cycle multiplier 610. Multi-cycle multiplier 610 computes T₂*y₂ which is output on line 650 to input “1” of MUX 620 with MUX 620 set to “1”. MUX 630 input is set to “0”. MUX 620 sends T₂*y₂ to input A of SAU 400 on line 670. Line 670 is directly connected to the minuend input of subtractor 710. Register memory 695 provides Y₁ on line 660 to input “0” of MUX 630 and MUX 630 is set to “0”. MUX 630 sends Y₁ to input B of SAU 400 on line 675. Line 675 is directly connected to the subtrahend input of subtractor 710. Subtractor 710 computes A−B (which is T₂*y₂−Y₁) and outputs A−B on line 730 to input “0” of MUX 740 with MUX 740 set to “0”. MUX 740 sends D (which is A−B) on line 690 to register memory 695 where it is stored in temporary register T₂.

Step 307 utilizes both multi-cycle multiplier 610 and SAU 400. Register memory 695 provides T₁ and x₂ on lines 635 and 640, respectively, to multi-cycle multiplier 610. Multi-cycle multiplier 610 computes T₁*x₂ which is output on line 650 to input “1” of MUX 620 with MUX 620 set to “1”. MUX 620 sends T₁*x₂ to input A of SAU 400 on line 670. Line 670 is directly connected to the minuend input of subtractor 710. Register memory 695 provides X₁ on line 660 to input “0” of MUX 630 and MUX 630 is set to “0”. MUX 630 sends X₁ to input B of SAU 400 on line 675. Line 675 is directly connected to the subtrahend input of subtractor 710. Subtractor 710 computes A−B (which is T₁*x₂−X₁) and outputs A−B on line 730 to input “0” of MUX 740 with MUX 740 set to “0”. MUX 740 sends D (which is A−B) on line 690 to register memory 695 where it is stored in temporary register T₁.

Steps 308-309 are performed in the microprocessor (not shown) without using multi-cycle multiplier 610 and SAU 400.

Step 310 utilizes multi-cycle multiplier 610. Register memory 695 provides T₁ on line 635 and Z₁ on line 640 to multi-cycle multiplier 610. Multi-cycle multiplier 610 computes T₁*Z₁ and the result is output on line 650 to register memory 695 where it is stored in temporary register T₂.

Step 311 utilizes multi-cycle multiplier 610. Register memory 695 provides T₁ on both lines 635 and 640 to multi-cycle multiplier 610. Multi-cycle multiplier 610 computes T₁ ² and the result is output on line 650 to register memory 695 Y₃ where it is stored in Y₃.

Step 312 utilizes multi-cycle multiplier 610. Register memory 695 provides T₂ on both lines 635 and 640 to multi-cycle multiplier 610. Multi-cycle multiplier 610 computes T₂ ² and the result is output on line 650 to register memory 695 where it is stored in X₃.

Step 313 utilizes multi-cycle multiplier 610. Register memory 695 provides T₁ on line 635 and Y₃ on line 640 to multi-cycle multiplier 610. Multi-cycle multiplier 610 computes T₁*Y₃ and the result is output on line 650 to register memory 695 where it is stored in temporary register T₁.

Step 314 utilizes both multi-cycle multiplier 610 and SAU 400. Register memory 695 provides X₃ on line 665 to input “0” of MUX 620 with MUX 620 set to “0”. MUX 620 sends X₃ to input A of SAU 400 on line 670. Line 670 is directly connected to the minuend input of subtractor 710. Register memory 695 provides T₁ on line 660 to input “0” of MUX 630 with MUX 630 set to “0”. MUX 630 sends T₁ on line 675 to input B of SAU 400. Line 650 is directly connected to the subtrahend input of subtractor 710. Subtractor 710 computes and outputs A−B (which is X₃−T₁) on line 730 to the minuend input of subtractor 720. Register memory 695 provides X₁ on line 635 and Y₃ on line 640 to multi-cycle multiplier 610. Multi-cycle multiplier 610 computes Y₃*X₁. The result is output on line 650 to input C of SAU 400 which is directly connected to logical one bit left shifter 715 which multiplies input C by two and outputs 2C (which is 2Y₃*X₁) on line 735 to the subtrahend output of subtractor 720. Subtractor 720 computes and outputs A−B−2C on line 750 to input “1” of MUX 740 with MUX 740 set to “1”. MUX 740 sends D (which is A−B−2C=X₃−T₁−2Y₃*X₁) on line 690 to register memory 695 where it is stored in X₃.

Step 315 utilizes both multi-cycle multiplier 610 and SAU 400. In step 314, Y₃*X₁ was computed by multi-cycle multiplier 610. Hence, Y₃*X₁ is still present in the output register (not shown) of multi-cycle multiplier 610 and in Step 315 is sent on line 650 to input “1” of MUX 620 and MUX 620 is set to “1”. MUX 620 sends Y₃*X₁ on line 670 to input A of SAU 400. Line 670 is connected directly to the minuend input of subtractor 710. Register memory 695 provides X₃ on line 660 to input “0” of MUX 630 with MUX 630 set to “0”. MUX 630 sends X₃ on line 675 to input B of SAU 400. Line 675 is directly connected to the subtrahend input of subtractor 710. Subtractor 710 calculates A−B and sends the result on line 730 to input “0” of MUX 740 with MUX 740 set to“0”. MUX 740 sends D (which is A−B=Y₃*X₁−X₃) on line 690 to register memory 695 which passes D through on line 635 and provides T₂ on line 640 to multi-cycle multiplier 610. Multi-cycle multiplier 610 computes D*T₂ (which is (Y₃*X₁−X₃)*T₂) and outputs the result on line 650 to register memory 695 where the result is stored in Y₃.

Step 316 utilizes both multi-cycle multiplier 610 and SAU 400. Register memory 695 provides Y₃ on line 665 to input “0” of MUX 620 with MUX 620 set to “0”. MUX 620 sends Y₃ on line 670 to input A of SAU 400. Line 670 is directly connected to the minuend of subtractor 710. Register memory 695 provides T₁ on line 635 and Y₁ on line 640 to multi-cycle multiplier 610. Multi-cycle multiplier 610 computes and outputs T₁*Y₁ on line 650 to input “1” of MUX 630 with MUX 630 set to “1”. MUX 630 sends T₁*Y₁ on line 675 to input B of SAU 400. Line 675 is directly connected to the subtrahend of subtractor 710. Subtractor 710 computes A−B (which is Y₃−T₁*Y₁) and provides the result on line 730 to input “0” of MUX 740 with MUX 740 set to “0”. MUX 740 sends D (which is Y₃−T₁*Y₁) on line 690 to register memory 695 where the result is stored in Y₃.

Step 317 returns the result of the addition of P+Q in Jacobian coordinates which is (X₃, Y₃, Z₃). 

1. An apparatus for performing an elliptic curve cryptography point addition operation using mixed affine-Jacobian coordinates comprising: a register memory for storing a first point in affine coordinates and a second point in Jacobian coordinates; a modular multiplier electrically coupled to the register memory; and a simple arithmetic processor electrically coupled to the register memory and the modular multiplier, wherein the simple arithmetic processor is configured to perform modular subtraction and modular multiplication by two in support of the point addition operation comprising a plurality of steps in mixed affine-Jacobean coordinates.
 2. The apparatus of claim 1 wherein the simple arithmetic processor comprises two modular subtractors.
 3. The apparatus of claim 1 wherein the simple arithmetic processor comprises a logical one bit left shifter.
 4. The apparatus of claim 1 wherein the simple arithmetic processor is configured to output A−B−2C for an input of variables A, B and C.
 5. The apparatus of claim 1 wherein the point addition operation is performed over a short Weierstrass curve of the form y=x³+ax+b where a=−3.
 6. The apparatus of claim 1 wherein the register memory is configured for two temporary storage variables, T₁ and T₂.
 7. A mobile device comprising the apparatus of claim
 1. 8. A smartcard comprising the apparatus of claim
 1. 9. The mobile device of claim 7 wherein the mobile device is a smartphone.
 10. The apparatus of claim 1 wherein the modular multiplier is configured to perform at most one modular multiplication for each one of the plurality of steps.
 11. A method for performing an elliptic curve cryptography point addition operation using mixed affine-Jacobian coordinates comprising: accepting the input of a first point in affine coordinates and a second point in Jacobian coordinates into a computational device having a register memory, a modular multiplier and a simple arithmetic processor configured for modular subtraction and modular multiplication by two; and enabling the computational device to execute a sequence of steps to perform the elliptic curve cryptography point addition operation of the first point and the second point wherein the modular multiplier performs at most one modular multiplication per step.
 12. The method of claim 11 wherein the simple arithmetic processor is configured to output A−B for an input of variables A and B.
 13. The method of claim 11 wherein the simple arithmetic processor is configured to output A−B−2C for an input of variables A, B and C.
 14. The method of claim 11 wherein the sequence of steps requires no more than two temporary variables.
 15. The method of claim 11 wherein the computational device is part of a mobile device.
 16. The method of claim 15 wherein the mobile device is a smartphone.
 17. Then method of claim 11 wherein the computational device is part of a smartcard.
 18. The method of claim 11 further comprising enabling the computational device to output a result of the point addition operation in Jacobian coordinates. 